Found problems: 85335
2019 Thailand TST, 2
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
2000 Harvard-MIT Mathematics Tournament, 7
A regular tetrahedron of volume $1$ is filled with water of total volume $\frac{7}{16}$. Is it possible that the center of the tetrahedron lies on the surface of the water? How about in a cube of volume $1$?
2006 Germany Team Selection Test, 1
Let $ ABC$ be an equilateral triangle, and $ P,Q,R$ three points in its interior satisfying
\[ \measuredangle PCA \equal{} \measuredangle CAR \equal{} 15^{\circ},\ \measuredangle RBC \equal{} \measuredangle BCQ \equal{} 20^{\circ},\ \measuredangle QAB \equal{} \measuredangle ABP \equal{} 25^{\circ}.\] Compute the angles of triangle $ PQR$.
2005 Iran MO (3rd Round), 2
Let $a\in\mathbb N$ and $m=a^2+a+1$. Find the number of $0\leq x\leq m$ that:\[x^3\equiv1(\mbox{mod}\ m)\]
2014 Taiwan TST Round 1, 2
Determine whether there exist ten sets $A_1$, $A_2$, $\dots$, $A_{10}$ such that
(i) each set is of the form $\{a,b,c\}$, where $a \in \{1,2,3\}$, $b \in \{4,5,6\}$, $c \in \{7,8,9\}$,
(ii) no two sets are the same,
(iii) if the ten sets are arranged in a circle $(A_1, A_2, \dots, A_{10})$, then any two adjacent sets have no common element, but any two non-adjacent sets intersect. (Note: $A_{10}$ is adjacent to $A_1$.)
2005 Germany Team Selection Test, 2
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.
[i]Proposed by Hojoo Lee, Korea[/i]
1979 IMO Longlists, 66
Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.
1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 10
The minimal value of $ f(x) \equal{} \sqrt{a^2 \plus{} x^2} \plus{} \sqrt{(x\minus{}b)^2 \plus{} c^2}$ is
A. $ a\plus{}b\plus{}c$
B. $ \sqrt{a^2 \plus{} (b \plus{} c)^2}$
C. $ \sqrt{b^2 \plus{} (a\plus{}c)^2}$
D. $ \sqrt{(a\plus{}b)^2 \plus{} c^2}$
E. None of these
1998 ITAMO, 6
We say that a function $f : N \to N$ is increasing if $f(n) < f(m)$ whenever $n < m$, multiplicative if $f(nm) = f(n)f(m)$ whenever $n$ and $m$ are coprime, and completely multiplicative if $f(nm) = f(n)f(m)$ for all $n,m$.
(a) Prove that if $f$ is increasing then $f(n) \ge n$ for each $n$.
(b) Prove that if $f$ is increasing and completely multiplicative and $f(2) = 2$, then $f(n) = n$ for all $n$.
(c) Does (b) remain true if the word ”completely” is omitted?
2004 Junior Balkan Team Selection Tests - Romania, 4
Consider a cube and let$ M, N$ be two of its vertices. Assign the number $1$ to these vertices and $0$ to the other six vertices. We are allowed to select a vertex and to increase with a unit the numbers assigned to the $3$ adjiacent vertices - call this a [i]movement[/i].
Prove that there is a sequence of [i]movements [/i] after which all the numbers assigned to the vertices of the cube became equal if and only if $MN$ is not a diagonal of a face of the cube.
Marius Ghergu, Dinu Serbanescu
2007 South East Mathematical Olympiad, 4
Let $a$,$b$,$c$ be positive real numbers satisfying $abc=1$. Prove that inequality $\dfrac{a^k}{a+b}+ \dfrac{b^k}{b+c}+\dfrac{c^k}{c+a}\ge \dfrac{3}{2}$ holds for all integer $k$ ($k \ge 2$).
V Soros Olympiad 1998 - 99 (Russia), 10.9
A triangle of area $1$ is cut out of paper. Prove that it can be bent along a straight segment so that the area of the resulting figure is less than $s_0$, where $s_0=\frac{\sqrt5-1}{2}$.
Note. The value $s_0$ specified in the condition can be reduced (the smallest value of$s_0$ is unknown to the authors of the problem). If you manage to do this (and justify it), write.
2024 Olimphíada, 1
Find all pairs of positive integers $(m,n)$ such that
$$lcm(1,2,\dots,n)=m!$$
where $lcm(1,2,\dots,n)$ is the smallest positive integer multiple of all $1,2,\dots n-1$ and $n$.
1990 India National Olympiad, 2
Determine all non-negative integral pairs $ (x, y)$ for which
\[ (xy \minus{} 7)^2 \equal{} x^2 \plus{} y^2.\]
2009 Romania Team Selection Test, 3
Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.
2024 Nepal TST, P1
Let $a, b$ be positive integers. Prove that if $a^b + b^a \equiv 3 \pmod{4}$, then either $a+1$ or $b+1$ can't be written as the sum of two integer squares.
[i](Proposed by Orestis Lignos, Greece)[/i]
2009 Bundeswettbewerb Mathematik, 1
Determine all possible digits $z$ for which
$\underbrace{9...9}_{100}z\underbrace{0...0}_{100}9$ is a square number.
2000 All-Russian Olympiad, 2
Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.
2009 Iran MO (3rd Round), 8
Sone of vertices of the infinite grid $\mathbb{Z}^{2}$ are missing. Let's take the remainder as a graph. Connect two edges of the graph if they are the same in one component and their other components have a difference equal to one. Call every connected component of this graph a [b]branch[/b].
Suppose that for every natural $n$ the number of missing vertices in the $(2n+1)\times(2n+1)$ square centered by the origin is less than $\frac{n}{2}$.
Prove that among the branches of the graph, exactly one has an infinite number of vertices.
Time allowed for this problem was 90 minutes.
2025 Philippine MO, P5
Find the largest real constant $k$ for which the inequality \[(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) + k(a - 1)(b - 1)(c - 1)(d - 1) \ge 0\] holds for all real numbers $a$, $b$, $c$, and $d$.
2018 Iran MO (1st Round), 5
There are $128$ numbered seats arranged around a circle in a palaestra. The first person to enter the place would sit on seat number $1$. Since a contagious disease is infecting the people of the city, each person who enters the palaestra would sit on a seat whose distance is the longest to the nearest occupied seat. If there are several such seats, the newly entered person would sit on the seat with the smallest number. What is the number of the seat on which the $39$th person sits?
2006 Singapore Senior Math Olympiad, 1
Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.
VMEO III 2006, 10.2
Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.
2008 Argentina National Olympiad, 4
Find all real numbers $ x$ which satisfy the following equation:
$ [2x]\plus{}[3x]\plus{}[7x]\equal{}2008$.
Note: $ [x]$ means the greatest integer less or equal than $ x$.
2017 Hong Kong TST, 2
In a committee there are $n$ members. Each pair of members are either friends or enemies. Each committee member has exactly three enemies. It is also known that for each committee member, an enemy of his friend is automatically his own enemy. Find all possible value(s) of $n$